1)
How does David know that Hashem accepted his supplication?
1.
Radak: He knew via Nevu'ah. We find that at the end of his days, when he was in danger from Yishbi b'Nov, David's men swore that he will not go with them to war again (Shmuel II, 21:16-17).
2.
Malbim: [Afterwards] David did not need to fight; he was helped via trust in Hashem.
